Answer
$\dfrac{1}{36}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To determine the number that will complete the square to solve the given equation, $
3w^2-w-24=0
,$ use first the properties of equality to express the equation in the form $x^2+bx=c.$ Once in this form, the needed number to complete the square of the left side is equal to $\left( \dfrac{b}{2} \right)^2.$
$\bf{\text{Solution Details:}}$
Using the properties of equality, in the form $x^2+bx=c,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{3w^2-w-24}{3}=\dfrac{0}{3}
\\\\
w^2-\dfrac{1}{3}w-8=0
\\\\
w^2-\dfrac{1}{3}w=8
.\end{array}
In the equation above, $b=
-\dfrac{1}{3}
.$ Using $\left( \dfrac{b}{2} \right)^2$, the number that will complete the square on the left side of the equal sign is
\begin{array}{l}\require{cancel}
\left( \dfrac{-\dfrac{1}{3}}{2} \right)^2
\\\\=
\left( -\dfrac{1}{3}\div2 \right)^2
\\\\=
\left( -\dfrac{1}{3}\cdot\dfrac{1}{2} \right)^2
\\\\=
\left( -\dfrac{1}{6} \right)^2
\\\\=
\dfrac{1}{36}
.\end{array}